Dental Mirror. A dentist uses a curved mirror to view teeth on the upper side of the mouth. Suppose she wants an erect with a magnification of 2.00 when the mirror is 1.25 cm from a tooth. (Treat this problem as though the object and lie along a straight line.) (b) What must be the focal length and radius of curvature of this mirror?

Verified step by step guidance

1. Identify the type of mirror used. Since the image is erect and magnified, the mirror is a concave mirror.
2. Use the mirror equation \(\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\) where \(f\) is the focal length, \(d_o\) is the object distance, and \(d_i\) is the image distance. Note that the object distance \(d_o\) is given as 1.25 cm.
3. Apply the magnification equation \(m = -\frac{d_i}{d_o}\) where \(m\) is the magnification. Since the magnification is given as 2.00 and the image is erect, \(d_i\) will be positive and twice the object distance.
4. Substitute the values of \(d_o\) and \(d_i\) into the mirror equation to solve for the focal length \(f\).
5. Once the focal length \(f\) is found, calculate the radius of curvature \(R\) using the relationship \(R = 2f\).

Key Concepts

Mirror Formula

The mirror formula relates the object distance (u), image distance (v), and focal length (f) of a mirror. It is expressed as 1/f = 1/v + 1/u. This formula is essential for determining the focal length of the mirror when the object and image distances are known, which is crucial for solving the problem regarding the dental mirror.

Magnification

Magnification (M) is the ratio of the height of the image (h') to the height of the object (h), and it can also be expressed in terms of distances as M = -v/u. In this scenario, a magnification of 2.00 indicates that the image appears twice as large as the object, which directly influences the calculations for the image distance and subsequently the focal length.

Radius of Curvature

The radius of curvature (R) of a mirror is related to its focal length (f) by the equation R = 2f. This relationship is important because once the focal length is determined using the mirror formula and magnification, the radius of curvature can be easily calculated, providing insight into the physical dimensions of the dental mirror.